Majority illusion is a phenomenon in social networks wherein the decision by the majority of the network is not the same as one's personal social circle's majority, leading to an incorrect perception of the majority in a large network. We present polynomial-time algorithms which completely eliminate majority illusion by altering as few connections in the network as possible. Eliminating majority illusion ensures each neighbourhood in the network has at least a 1/2-fraction of the majority winner. This result is surprising as partially eliminating majority illusion is NP-hard. We generalize the majority illusion problem to an arbitrary fraction p and show that the problem of ensuring all neighbourhoods in the network contain at least a p-fraction of nodes consistent with a given preference is NP-hard, for nearly all values of p.

In the famous network creation game of Fabrikant et al. a set of agents play a game to build a connected graph. The $n$ agents form the vertex set $V$ of the graph and each vertex $v\in V$ buys a set $E_v$ of edges inducing a graph $G=(V,\bigcup\limits_{v\in V} E_v)$. The private objective of each vertex is to minimize the sum of its building cost (the cost of the edges it buys) plus its connection cost (the total distance from itself to every other vertex). Given a cost of $α$ for each individual edge, a long-standing conjecture, called the tree conjecture, states that if $α> n$ then every Nash equilibrium graph in the game is a spanning tree. After a plethora of work, it is known that the conjecture holds for any $α>3n-3$. In this paper we prove the tree conjecture holds for $α>2n$. This reduces by half the open range for $α$ with only $[n, 2n)$ remaining in order to settle the conjecture.

Abstract. We present a [Formula: see text]-approximation algorithm for the matching augmentation problem (MAP): given a multigraph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A [Formula: see text]-approximation algorithm for the same problem was presented recently; see Cheriyan et al. [ Math. Program., 182 (2020), pp. 315–354]. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems.

We present a $\frac53$-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A $\frac74$-approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a $\frac{7}{4}$-approximation algorithm," {\em Math. Program.}, 182(1):315--354, 2020; arXiv:1810.07816. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems.

We study the fair division of a collection of mindivisible goods amongst a set of nagents. Whilst envy-free allocations typically do not exist in the indivisible-goods setting, envy-freeness can be achieved if some amount of a divisible good (money) is introduced. Specifically, Halpern and Shah (SAGT 2019, pp.374-389) showed that, given additive valuation functions where the marginal value of each good is at most one dollar for each agent, there always exists an envy-free allocation requiring a subsidy of at most (n-1)·m dollars. The authors also conjectured that a subsidy of $n-1$ dollars is sufficient for additive valuations. We prove this conjecture. In fact, a subsidy of at most one dollar per agent is sufficient to guarantee the existence of an envy-free allocation. Further, we prove that for general monotonic valuation functions an envy-free allocation always exists with a subsidy of at most 2(n-1) dollars per agent. In particular, the total subsidy required for monotonic valuations is independent of the number of goods.